How can I manually calculate the output a specific neural network given some input?
Given that the neurons are linear in the hidden layer of the neural network, so the output is just the dot product of the weights and the input. To put things in perspective, generally, we use an activation function (sigmoid, signum, etc.), which is applied to the dot product.

Hence, for an input of $3$ inputs to node 1 of hidden layer is $-3 * 3 = -9$, to node 2 is $2 * 3= 6$ to node 3 is $-1 * 3 = -3$ to node 4 is $0.5 *3 = 1.5$ (basically, I have performed dot product). Since neurons are linear, no activation function is applied and is directly propagated to the next layer, the output layer.

Contribution of all the hidden nodes to the output layer is the dot product of the weights with the output of the hidden layer that is $9*2 + -0.5*6 + -3*-3 + 1*1.5 = -10.5$ (check for any calculation errors). And, finally, the output layer is bipolar hence an activation function (probably sigmoid) is applied to it. Since for sigmoid function is smaller than $0.5$ if $z&lt;0$, hence the output will be $0$, as $z = -4.5$, as per convention.